Are all of them proper but describe different sets/arragement/anything of paper?
What are meanings, differences, which are proper?

In the Free Dictionary, I find cyclic definitions for
stack,
pile,
heap:

stack → An orderly pile, especially one arranged in layers. See Synonyms at heap.
stacking → To arrange in a stack; pile.
pile → A quantity of objects stacked or thrown together in a heap. See Synonyms at heap.
piling → To form a heap or pile.
heap → To put or throw in a pile.

I tend to think that they imply different levels of orderlyness. A 'heap' is the least orderly collection of papers. A pile is a bit more tidy, and a stack is a fairly neat arrangement of paper.
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JimAug 27 '12 at 20:18

As a CS+IT person I have never heard of "a heap of papers" being used at all in my profession. The only time I've heard it used with regards to paper is in the context of "the papers lay in a heap", indicating a completely disordered pile of items.
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Roddy of the Frozen PeasAug 27 '12 at 22:54

2 Answers
2

I feel for your confusion at the circular definitions. Here is how I think of these as a native English speaker.

First, you know that all of these terms refer to many things all located close to each other so they are thought of as related as a group. Usually, if they are physical things, they will be very close, maybe even touching each other, often on top of each other. If they are abstract things, then they are related in another sense of close together or collected.

Usually, these terms are not used for live or animate things, unless to make a point by being odd. However, they can be used for dead things, e.g., a pile of corpses.

Now, there are two properties that you have to ask about to tell the terms apart.

How big is the group?

How organized is the group?

A heap is big. A stack and pile can be small, medium, or big.

A stack is organized. A pile and heap can be organized or unorganized, but often unorganized. Other concepts similar to organization here are alignment and order.

These are only the main meanings that come to mind. It would not be incorrect to say that an organized group of papers is a pile of papers. It would also not be wrong to tell someone to organize a stack of papers, because maybe it is not organized to this person's standards.

So if a group of papers is small, you should not call it a heap. And if a group of papers is very messy, you should not call it a stack. The others are more flexible and vague.

When you are talking about abstract objects, you should be more careful about using stack and pile. It is common to say he is in a heap of trouble, I have a heap of problems, or I'm gonna/going to give him a heaping of hurt. The word mound also covers some of the same ground. It is less common to describe a group of abstract things as a pile or stack. Two notable exceptions are of course when these are technical terms for mathematical objects or datatypes. I suspect this might be because the real meaning that heap adds in the former case is bigness. I have a heap of problems means that you have a lot of problems all close together.

If this is a lot to remember at once, you can think roughly of these terms as growing increasingly specific. RoaringFish's comment made me think of this ordering, which is by increasing messiness:

stack ≤ pile ≤ heap

You might also think of these as being proper inclusions if you are familiar with sets:

stack ⊂ pile ⊂ heap

This picture depicts these relationships.

This means that everything that counts as a stack can also be called a pile or a heap, though calling it one of these is being less specific (you should generally be as specific as appropriate). Everything that can be called a pile can also be called a heap. But you cannot go in the other direction. There are things that you can call heaps but cannot call stacks.

Thank you Rachel. Thanks to your answer I understand much more also in other contexts than group of papers. I make notes from answer :).
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Grzegorz WierzowieckiAug 27 '12 at 22:30

Even answer is great, I will wait with selecting as "solution" to give others chance.
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Grzegorz WierzowieckiAug 27 '12 at 22:31

@Rachel I'm glad you referenced that 'heap' implies 'big size or quantity.' Anecdotally, I think of the use of 'heap' only among my family in the U.S. Southeast. Does this implication of size apply to other regions? Also, I would not use 'heap' in any formal setting.
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MikeAug 28 '12 at 13:23

@Mike: I am from Tampa, Florida, which I don't think of as part of The South culturally, but who knows. I don't say heap much, but I can't recall it getting a strange reaction from anyone anywhere. I think heaping spoonful is possibly common outside of the southeast. I've heard and read it many times.
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RachelAug 29 '12 at 2:30

I think part of the difficulty here arises because usage has changed. See what google NGram makes of heap / pile of sand: books.google.com/ngrams/…
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PitarouAug 29 '12 at 18:51

A stack is arranged one on top of the other. A stack of papers would be arranged in a vertical orientation. Stacks are also countable in this sense, as you can have two stacks of papers sitting directly next to one another. A stack is about the arrangement.

Heap and pile are fairly interchangeable, with a slight nuance between them. (Some may disagree, as this is the best I can describe it), a heap carries more of an implication that the papers were collected into a group with an intention of stockpiling them. A pile reminds me of a group of objects that were collected with more reckless abandon, or with no intention behind it at all--they just happen to be gathered together in a group for whatever reason.

Either way, a big difference is that if you had two heaps/piles in close proximity to one another, a native speaker would be likely to consider them a single pile, and refer to them in the singular "a pile/heap". For stacks, on the other hand, when there are many of them in close proximity, they are distinctly considered, and we are more likely to refer to them as "two stacks, three stacks, etc."

I think that pile is a general word that encompasses both heap and stack, so your stack of papers could also be a heap of papers. Is this a difference in dialect? (I speak British English.)
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PitarouAug 28 '12 at 3:25

2

@Pitarou: Personally, I think stack is more specific/exclusive. Any stack can be considered a heap, but not every heap would accurately be a stack. Also, when you are being more vague, such as when talking about abstract things, it makes more sense to say "a heap of trouble/problems/etc." than "a stack of trouble/problems/etc.". I think pile is a lot like heap.
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RachelAug 28 '12 at 3:38

@Rachel I think mentioning abstract things like "a heap of ideas/believes/trouble/problems/etc" is very interesting itself - I would suggest adding it to your answer as "P.S." , bur only if agree that might be a good idea, without shadowing main answer message.
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Grzegorz WierzowieckiAug 28 '12 at 9:31

@Pitarou to expand on Rachel's comment, you could definitely call a stack a pile/heap in American English. But it's all a matter of nuance and implication. If you care about the arrangement, you'll call it a stack. I don't know much about British English, but I'd be surprised if it were different in this respect.
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AtaraxiaAug 28 '12 at 15:29

@GrzegorzWierzowiecki: I mentioned it in my answer originally. I suppose I could say more about it.
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RachelAug 28 '12 at 21:58